ILC WITH ADAPTIVE WEIGHTING PARAMETERS IN THE OBJECTIVE FUNCTION

7.1 Introduction

In this chapter a study on iterative learning control with adaptive weighting parameters in the objective function is presented. This method requires clever solution of the linear quadratic optimisation problem to improve convergence speed and ensuring stability. Attempts to achieve very fast convergence typically cause instability and vice- versa. Both this control criteria is determined by minimization/maximization a quadratic objective function. In a discrete control system, stability is typically related to convergence rate.

A typical quadratic objection function is as given below:

] ~ ~ [ 2 1 min _{1 1 1 1} 1 1         k T k k T k k k J e Qe U R U U    (7.1)

The first term in the cost criterion is the variance and the second one is the input change need to minimize the system variance while improving the system convergence. In ILC based on linear quadratic optimal control, the weighting matrices Q and R are important design parameters (Amann et al., 1996). The weighting matrices Q and R are used to maintain an optimal ratio between control policy change and error minimization. Convergence rate depends on the ratio of the weighting parameters in the quadratic objective function. It is essential that the control policy does not deviate too much from that of the previous batch (the kth batch) whilst reducing tracking error in order to maintain stability (Amann et al., 1996). In this study, optimisation based on linearised models is used to calculate the control policy for the new batch. As for the batch-to-batch control policy, the control policy for every successive batch should be reducing the tracking error in product quality.

In Chapters 4, 5 and 6, the Q and R ratio in quadratic objective function were kept constant for all the batch runs with ILC. The weighting parameters used are Q is 1 and R is 0.0001. There have been significant improvements in batch to batch product concentrations when different techniques were introduced. However, there were some performance patterns that revealed high biomass concentration for the first few batches and then deteriorated for the rest of the batches forming a dumbbell curve as seen in

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Chapter 5, Figure 5.5 for λ=10. In Chapter 4, Figure 4.12(b), the batch to batch product improvement for Q=1 and R=0.00001I exhibited very good convergence rate for the first four batches and then deteriorated for the following batches. If only these performances can be sustained or further improved, the desired trajectory can be achieved asymptotically much faster within smaller number of batches. Bearing this in mind, an idea to adjust the Q and R ratio from batch to batch was developed. Instead of using fixed Q and R ratio, an adaptive ratio may improve the convergence rate without compensating the system stability. The question here is as to how to systematically adjust the Q and R ratio from batch to batch.

It is known that ratio of Q and R in the objective function determines the convergence rate of biomass concentration towards desired trajectory. For batch-to-batch ILC, stability is closely related to convergence. A converging performances means the system is steadily improving from batch to batch. Having a control over the convergence speed of the system to be controlled will be beneficial. Amann et al. (1996) developed a strategy to control the convergence speed by manipulating the weighting parameters, Q and R in the objective function to achieve desired convergence rate. An optimization principle using Ricatti feedback in combination with conventional feed-forward ILC was introduced. A performance criterion which evaluates both current run feed-back mechanism and feed-forward of previous trial data was developed to be used to tune the quadratic objective function.

Amann et al. (1996) suggested that R0 is fixed and then it is defined by the

objective function through R=ρR0. The weighting parameter R is determined

automatically. The scalar, ρ is a variable. The ρ is derived from equation σ2= σo2/ρ where

σ is the smallest singular value of gain (G) and σo is the smallest singular value

corresponding to R0. The smaller the ρ is, the faster the convergence will be. Hence, the

parameter ρ can be used to control the convergence rate of a controlled system. The study was tested on linear, continuous steady state. The proposed method improved convergence rate for tracking error and input sequence trajectory. It was used for time- variant dimension within a trial. The detailed study revealed that determination of control policy in the same manner for batch-to-batch control was not delivering similar performance.

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The advantage of the automated Q and R ratio used in constructing control policy is that we have control over convergence speed. Drawback is the magnitude of feed-forward is highly dependent on the state variance. Therefore, in worst case scenario, a sudden hike in input control policy may exist. The author did not consider the non- linear case, presence of any kind of disturbances and robustness of the study. Complete automated ratio tuning may not deliver stable performance for non-linear systems, in the presence of disturbance and/or model-plant mismatches.

Gao et al. (2001) diversified from Amann et al. (1996) by applying Q and R ratio modulating idea to general batch process. This study was done on a non-linear actuator process. Finite time interval is present and not automatically decided. In Amann et al., 1996 the batch run time is decided upon attainment of the desired performance which is decided by the automated Q and R ratio tuning. The effect of initialization error and unknown disturbance were taken into consideration in the work by Gao et al., 2001. The work presented by Gao et al. (2001) was on exponentially reducing the Q and R ratio, trial after trial. The ratio of Q and R was first fixed and then it was reduced trial after trial in the order of 0.6k-1. In this work ρ was the fixed Q and R ratio, which is 0.6. The ρ is to approach zero with increasing cycle number, k. The experimental results without the proposed method demonstrated that accumulation of initialization error and unknown disturbances causes instability in constant Q and R ratio. The exponentially reducing ρ after every batch run exhibits desirable convergence whilst minimizing tracking error and suppressing unknown disturbances and initialization errors.

This work is an adaptation of the works presented by Amann et al. (1996) and Gao et al. (2001). Both the introduction of ρ and systematically reducing Q and R ratio has been incorporated into this study. Fermentation process involves life mechanism and so it may not be that easy to automate the magnitude of feed rate. It is important that feed does not sway too high compared to previous feed rate to prevent cell damage. A more controlled adaptive Q and R ratio scheme is introduced in this chapter. The simulation results are discussed for PCR, PLS and MLR updated models for batch to batch iterative learning control.

This chapter is organised as follows. Section 7.2 presents the details of the proposed method for this chapter. Two methods are proposed in this section. One is the continuously reducing Q and R ratio from trial to trial and the other is error adaptive Q

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and R ratio. Section 7.3 presents the results for batch to batch continuously decreasing Q and R ratio for batch to batch ILC with updated linearised MLR, PCR and PLS models. Two different R0 values were tested for this method. Section 7.4 presents the results for

error adaptive Q and R ratio applied in batch to batch ILC with updated linearised MLR, PCR and PLS models. This method was also tested with 2 different R0 values. Section

7.5 presents the results for combination of model prediction confidence bounds (the method proposed in Chapter 5) and continuously decreasing Q and R ratio technique. The combined method was applied to MLR, PCR and PLS models. Section 7.6 concludes the chapter.

7.2 The proposed method

There is no proper guidance on the selection of Q and R. It is the ratio of Q and

R rather than the absolute value of these weighting matrices that matters in determining

the convergence speed. Large ratio leads to unstable system due to insensitivity of the systems variance. A strong feed-forward action tends to accumulate process uncertainties causing strong fluctuation in control policy. Very small ratio causes slow convergence and is undesirable (Gao et al., 2001).

Multi-objective optimisation problem does not have a straight forward solution (Tousain et al, 2001). Identification of Q and R can be made simpler by fixing the Q to Q=1 and then identifying the appropriate R to achieve desired convergence rate as well as minimize tracking error, ensuring stability and robustness by eliminating the effect of disturbances (Rogers, 2008; Tousain et al, 2001). R can be used to tune the performance of the learning controller (Phan, 1998; Tousain et al, 2001). Therefore, Q is usually fixed as 1 and R is identified by trial and error as discussed in Chapter 4. In Chapter 4, the R value was selected firstly based on the stability of the control system and then the convergence rate was considered. The R value selected for study in the previous chapters is the one with convergence rate compensated for stability. Actually, much smaller R value produced higher convergence rate and desired trajectory is attainable within 3 to 4 batch runs without disturbances. However, the constant high feed-forward signal did not provide a stable outcome to sustain the good performance.

When there is no disturbance, a constant and big ratio gives very good convergence rate. When there are disturbances, the ratio should be smaller to ensure stability. In addition to that, when the desired trajectory is almost achieved, there is no

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need to improve the feed so much and so reduced ratio would be good enough. Therefore, the ratio adapting to the error is suggested. It does not have to be reducing all the way through for all the batches as suggested by Gao et al. (2001). When there is disturbance, the ratio just has to be smaller and reducing till a second stable platform is found. When it is stable, the smaller ratio can be made constant. It is necessary that varying weighting schemes are applied to cater to practical application possibilities (Gao et al., 2001).

In this study, the initial Q and R values were fixed as Q0=1 and R0=0.0001I or

R0=0.00001I. The value R=0.0001I has been used throughout the study and so it will be

used as R0 for this technique. The R value of 0.00001I was dismissed in Chapter 4 due to

instability in performance. It will be interesting to evaluate the performance of this value with the adaptive technique. The other R0 value that was tested for the proposed method

is 0.00001I. The Q and R ratio is then reduced in accordance to a pre-specified weighted